Consider the following $3 \times 2$ array formed by using the numbers $1,2,3,4,5,6$:
$$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ a_{31} & a_{32} \end{bmatrix} = \begin{bmatrix} 1 & 6\\ 2 & 5\\ 3 & 4\end{bmatrix}$$
Observe that all row sums are equal, but the sum of the squares is not the same for each row. Extend the above array to a $3 × k$ array ${(a_{ij})_{3×k}}$ for a suitable k, adding more columns, using the numbers $7,8,9,..., 3k$ such that
$$\sum_{j=1}^k a_{1i}=\sum_{j=1}^k a_{2i}=\sum_{j=1}^k a_{3i}$$
$$and$$
$$\sum_{j=1}^k a_{1i}^2 = \sum_{j=1}^k a_{2i}^2 = \sum_{j=1}^k a_{3i}^2$$
I basically know its answer now. This was an olympiad problem and I did proceed in the right direction to solve it but I don't know what actually happened in the last step(s). I just got a small silly equation which did not give any solution to the question.